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Simplifying 3x2 = x2 + 6x + 24 Reorder the terms: 3x2 = 24 + 6x + x2 Solving 3x2 = 24 + 6x + x2 Solving for variable 'x'. Reorder the terms: -24 + -6x + 3x2 + -1x2 = 24 + 6x + x2 + -24 + -6x + -1x2 Combine like terms: 3x2 + -1x2 = 2x2 -24 + -6x + 2x2 = 24 + 6x + x2 + -24 + -6x + -1x2 Reorder the terms: -24 + -6x + 2x2 = 24 + -24 + 6x + -6x + x2 + -1x2 Combine like terms: 24 + -24 = 0 -24 + -6x + 2x2 = 0 + 6x + -6x + x2 + -1x2 -24 + -6x + 2x2 = 6x + -6x + x2 + -1x2 Combine like terms: 6x + -6x = 0 -24 + -6x + 2x2 = 0 + x2 + -1x2 -24 + -6x + 2x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -24 + -6x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -3x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -3x + x2)' equal to zero and attempt to solve: Simplifying -12 + -3x + x2 = 0 Solving -12 + -3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -3x + 12 + x2 = 0 + 12 Reorder the terms: -12 + 12 + -3x + x2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -3x + x2 = 0 + 12 -3x + x2 = 0 + 12 Combine like terms: 0 + 12 = 12 -3x + x2 = 12 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 12 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 12 + 2.25 Combine like terms: 12 + 2.25 = 14.25 2.25 + -3x + x2 = 14.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (x + -1.5) equal to 3.774917218 and -3.774917218.Subproblem 1
x + -1.5 = 3.774917218 Simplifying x + -1.5 = 3.774917218 Reorder the terms: -1.5 + x = 3.774917218 Solving -1.5 + x = 3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.774917218 + 1.5 x = 3.774917218 + 1.5 Combine like terms: 3.774917218 + 1.5 = 5.274917218 x = 5.274917218 Simplifying x = 5.274917218Subproblem 2
x + -1.5 = -3.774917218 Simplifying x + -1.5 = -3.774917218 Reorder the terms: -1.5 + x = -3.774917218 Solving -1.5 + x = -3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.774917218 + 1.5 x = -3.774917218 + 1.5 Combine like terms: -3.774917218 + 1.5 = -2.274917218 x = -2.274917218 Simplifying x = -2.274917218Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.274917218, -2.274917218}Solution
x = {5.274917218, -2.274917218}
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